Electricity and Magnetism: Key Points Review | 电与磁：考点精讲

📚 Electricity and Magnetism: Key Points Review | 电与磁：考点精讲

Electricity and magnetism lie at the heart of both IB and CIE physics specifications. Mastering these interconnected topics requires more than memorising formulae — it demands a clear conceptual understanding of fields, circuits, induction and their applications. This revision guide breaks down every essential point you need to know, from electric charge and Coulomb’s law to transformers and electromagnetic induction. Let’s step into this powerful duo of physics.

电与磁是 IB 和 CIE 物理课程的核心模块。掌握这些相互关联的知识点不单是背诵公式，更需要透彻理解场、电路、感应以及它们的实际应用。这篇考点精讲为你拆解从电荷、库仑定律到变压器与电磁感应的每一个必考要点。让我们一起走进这套强大的物理组合。

1. Electric Charge and Conservation | 电荷与守恒定律

Charge is a fundamental property of matter that exists in two types: positive and negative. Like charges repel, while unlike charges attract. The SI unit of charge is the coulomb (C). The smallest unit of free charge is the elementary charge e = 1.60 × 10⁻¹⁹ C; protons carry +e and electrons carry –e.

电荷是物质的基本属性，分正电荷和负电荷两种。同种电荷相互排斥，异种电荷相互吸引。电荷的国际单位是库仑（C）。自然界中最小的自由电荷单位是元电荷 e = 1.60 × 10⁻¹⁹ C，质子带 +e，电子带 –e。

The law of conservation of charge states that the total electric charge in an isolated system remains constant. Charge cannot be created or destroyed, only transferred from one object to another. This principle is crucial when analysing charging by friction, conduction and induction.

电荷守恒定律指出，孤立系统的总电荷量始终保持不变。电荷不能被创造或消灭，只能从一个物体转移到另一个物体。在分析摩擦起电、接触起电和感应起电时，这条定律是重要的判断依据。

2. Coulomb’s Law and Electric Force | 库仑定律与电场力

Coulomb’s law quantifies the force between two point charges: it is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. In equation form:

库仑定律定量描述了两个点电荷之间的作用力：力与电荷量的乘积成正比，与它们之间距离的平方成反比。其公式为：

F = k q₁ q₂ / r²

where k = 8.99 × 10⁹ N m² C⁻² is Coulomb’s constant. The force acts along the line joining the centres of the two charges. If charges have the same sign, the force is repulsive; if opposite, it is attractive. This force is analogous to Newton’s law of gravitation but can be both attractive and repulsive.

其中 k = 8.99 × 10⁹ N m² C⁻² 是库仑常数。力的方向沿着两电荷的连线。若电荷同号，力为排斥力；若异号，则为吸引力。库仑力与万有引力定律形式相似，但既可以是引力也可以是斥力。

In any electric force calculation, the magnitude is always positive, but direction must be determined from the types of charges involved. For multiple charges, the net force on a given charge is the vector sum of the individual forces — use superposition.

在计算电场力时，力的大小总取正值，但方向需根据电荷的电性判断。对于多个电荷，作用在某电荷上的合力是各个力的矢量和，即使用叠加原理。

3. Electric Fields and Field Lines | 电场与电场线

An electric field is a region of space where a stationary charge experiences a force. The electric field strength E is defined as the force per unit positive charge: E = F / q, with units N C⁻¹ or V m⁻¹. Field direction is the direction of the force on a positive test charge.

电场是静止电荷会受到力的空间区域。电场强度 E 定义为单位正电荷所受的力：E = F / q，单位为 N C⁻¹ 或 V m⁻¹。电场的方向是正试探电荷受力的方向。

Electric field lines represent the field visually: they start on positive charges and end on negative charges, never cross, and their density indicates field strength. For a point charge Q, the field a distance r away is E = k Q / r². Between two oppositely charged parallel plates, the field is uniform and its magnitude is E = V / d, where V is the potential difference and d is the plate separation.

电场线能直观表示电场：它从正电荷出发，终止于负电荷；电场线永不相交，其疏密反映场强大小。对于点电荷 Q，距离 r 处的电场大小为 E = k Q / r²。在两块带异种电荷的平行板之间，电场是匀强的，大小为 E = V / d，其中 V 是电势差，d 是板间距。

A charge q placed in a uniform electric field experiences a constant force F = qE. This is used to deflect electron beams or to accelerate charged particles.

将电荷 q 放在匀强电场中会受到恒力 F = qE。这一原理常用于偏转电子束或加速带电粒子。

4. Current, Potential Difference and Resistance | 电流、电势差与电阻

Electric current I is the rate of flow of charge: I = ΔQ / Δt. The SI unit is the ampere (A), where 1 A = 1 C s⁻¹. In metallic conductors, current is carried by moving electrons, while in electrolytes, both positive and negative ions contribute.

电流 I 是电荷流动的速率：I = ΔQ / Δt，国际单位为安培（A），1 A = 1 C s⁻¹。在金属导体中，电流由运动电子承载；在电解液中，正、负离子均参与导电。

Potential difference (p.d.) V between two points is the work done per unit charge to move charge between them: V = W / Q, measured in volts (V). The electromotive force (e.m.f.) ε of a source is the energy supplied per unit charge.

两点间的电势差 V 是移动单位电荷所做的功：V = W / Q，单位为伏特（V）。电源的电动势 ε 是电源每提供单位电荷所给的能量。

Ohm’s law states that for many conductors at constant temperature, the current through them is directly proportional to the p.d. across them, giving a constant resistance: V = IR. Resistance is measured in ohms (Ω). A component that obeys Ohm’s law is said to be ohmic. Filament lamps and diodes are non‑ohmic.

欧姆定律指出，许多导体在温度恒定时，通过它的电流与两端电压成正比，此时电阻为定值：V = IR。电阻的单位是欧姆（Ω）。遵循欧姆定律的元件称为欧姆导体。灯丝灯泡和二极管则是非欧姆导体。

Resistance depends on the material and geometry: R = ρ L / A, where ρ is the resistivity of the material, L is the length and A is the cross‑sectional area. Power dissipated in a resistor is P = VI = I²R = V²/R.

电阻取决于材料与几何形状：R = ρ L / A，其中 ρ 是材料的电阻率，L 是长度，A 是截面积。电阻上消耗的功率为 P = VI = I²R = V²/R。

5. DC Circuits and Kirchhoff’s Laws | 直流电路与基尔霍夫定律

In series circuits, the current is the same through all components. The sum of the p.d.s across each component equals the source p.d. Total resistance is Rₜ = R₁ + R₂ + … . In parallel circuits, the p.d. across each branch is the same. The total current is the sum of the branch currents. Total resistance is given by 1/Rₜ = 1/R₁ + 1/R₂ + … .

串联电路中，通过每个元件的电流相等。各元件两端的电压之和等于电源电压。总电阻为 Rₜ = R₁ + R₂ + … 。并联电路中，各支路两端电压相等。总电流等于各支路电流之和。总电阻满足 1/Rₜ = 1/R₁ + 1/R₂ + … 。

Kirchhoff’s current law (KCL): at any junction, the total current entering equals the total current leaving (ΣIₐᵢ = ΣIₒᵤₜ). Kirchhoff’s voltage law (KVL): around any closed loop, the algebraic sum of the e.m.f.s and p.d.s is zero (Σε + Σ(IR) = 0). These laws are essential for analysing complex circuits with multiple loops.

基尔霍夫电流定律（KCL）：流入任意节点的电流之和等于流出电流之和（ΣI入 = ΣI出）。基尔霍夫电压定律（KVL）：沿任一闭合回路，电动势与电势降的代数和为零（Σε + Σ(IR) = 0）。这两条定律是分析多回路复杂电路的基本工具。

Real cells have internal resistance r. The terminal p.d. V is less than the e.m.f. ε when current I flows: V = ε – Ir. Maximum power is delivered to a load when the load resistance equals the internal resistance.

实际电池具有内阻 r。当电流 I 流过时，端电压 V 小于电动势 ε：V = ε – Ir。当外接负载电阻等于内阻时，电池向负载输出的功率最大。

6. Magnetism and Magnetic Fields | 磁性与磁场

Magnets have north (N) and south (S) poles. Like poles repel, unlike poles attract. Magnetic field lines emerge from the N‑pole and enter the S‑pole externally, forming closed loops. The density of field lines indicates magnetic flux density B, measured in tesla (T).

磁体具有北极（N）和南极（S）。同极相斥，异极相吸。磁感线在外部从 N 极出发进入 S 极，形成闭合曲线。磁感线密度表示磁通量密度 B，单位是特斯拉（T）。

An electric current creates a magnetic field, as first shown by Ørsted. The direction of the circular field around a straight wire is given by the right‑hand grip rule: thumb points in current direction, fingers curl in field direction. For a solenoid, the field inside is strong and uniform, resembling that of a bar magnet. The right‑hand grip rule for a solenoid: fingers curl in the direction of conventional current, thumb points to the N‑pole.

电流产生磁场，这由奥斯特实验首次揭示。通电直导线周围的磁场呈环形，方向可用右手螺旋定则判断：拇指指向电流方向，四指弯曲方向即磁场方向。对于螺线管，内部磁场强且均匀，类似条形磁铁。右手握住螺线管，四指沿电流方向，拇指所指即为 N 极。

Magnetic flux Φ passing through an area A is defined as Φ = B A cos θ, where θ is the angle between the field and the normal to the area. Unit: weber (Wb). Magnetic flux linkage for a coil of N turns is NΦ.

穿过面积 A 的磁通量 Φ 定义为 Φ = B A cos θ，θ 是磁场方向与面积法线的夹角，单位韦伯（Wb）。对于 N 匝线圈，磁链为 NΦ。

7. Magnetic Force on a Current-Carrying Conductor | 通电导体在磁场中的力

A conductor carrying current in a magnetic field experiences a force if the current is not parallel to the field. The magnitude is F = B I L sin θ, where θ is the angle between the current direction and B. Maximum force occurs when the conductor is perpendicular to the field (θ = 90°).

通电导体在磁场中，若电流方向不与磁场平行，则会受到力的作用。力的大小为 F = B I L sin θ，θ 是电流方向与 B 的夹角。当导体与磁场垂直（θ = 90°）时，力最大。

The direction of the force is given by Fleming’s left‑hand rule: thumb – force (motion), first finger – field (N to S), second finger – current (conventional direction, + to –). This force is the principle behind electric motors.

力的方向用弗莱明左手定则判断：拇指—力（运动）方向，食指—磁场方向（N→S），中指—电流方向（+→–）。该力是电动机的工作原理。

For a rectangular coil of N turns in a magnetic field, the torque (turning effect) is τ = N B I A sin φ, where A is the coil area and φ is the angle between the field and the normal to the coil. A split‑ring commutator reverses the current each half‑turn to keep the coil rotating in one direction.

对于一个面积为 A 的 N 匝矩形线圈，在磁场中所受力矩为 τ = N B I A sin φ，φ 是磁场与线圈法线的夹角。换向器（整流子）每半圈改变电流方向，使线圈连续向同一方向旋转。

8. Electromagnetic Induction and Faraday’s Law | 电磁感应与法拉第定律

Electromagnetic induction is the production of an e.m.f. (induced e.m.f.) across a conductor when it is exposed to a changing magnetic flux. The size of the induced e.m.f. is directly proportional to the rate of change of magnetic flux linkage:

电磁感应是指当导体处在变化的磁通量中时，导体两端产生感应电动势的现象。感应电动势的大小与磁链的变化率成正比：

ε = – N ΔΦ / Δt

The negative sign represents Lenz’s law. An induced e.m.f. can be created by moving a magnet into a coil, changing the coil area, rotating the coil in a magnetic field, or changing the current in an electromagnet.

式中的负号体现了楞次定律。将磁铁插入线圈、改变线圈面积、在磁场中旋转线圈，或改变电磁铁中的电流，都可以产生感应电动势。

Lenz’s law states that the direction of the induced current is such that it opposes the change in magnetic flux that produced it. This is a consequence of conservation of energy — if the induced current aided the change, energy would be created from nothing.

楞次定律指出，感应电流的方向总是使其产生的作用阻碍引起感应电流的磁通量变化。这是能量守恒的体现——如果感应电流促进变化，就会无中生有地创生能量。

For a straight conductor of length L moving with velocity v perpendicular to a magnetic field B, the induced e.m.f. is ε = B L v. This is a special case of Faraday’s law and is often tested in the context of a rod sliding on rails.

对于长度为 L 的直导体，以速度 v 垂直于磁场 B 运动时，感应电动势为 ε = B L v。这是法拉第定律的特殊情况，常见于导轨与滑杆问题。

9. Generators, AC and Transformers | 发电机、交流电与变压器

An a.c. generator (alternator) works by rotating a coil in a uniform magnetic field. The coil is connected to an external circuit via slip rings and brushes, producing an alternating e.m.f. that varies sinusoidally with time. The peak e.m.f. is ε₀ = N B A ω, where ω is the angular frequency.

交流发电机通过线圈在匀强磁场中旋转工作。线圈通过滑环和电刷与外部电路相连，产生随时间按正弦规律变化的交变电动势。峰值电动势为 ε₀ = N B A ω，ω 是角频率。

For an ideal transformer, the ratio of primary voltage to secondary voltage equals the ratio of turns: Vₚ / Vₛ = Nₚ / Nₛ. Assuming 100% efficiency, input power equals output power: Vₚ Iₚ = Vₛ Iₛ, so Iₚ / Iₛ = Nₛ / Nₚ. Step‑up transformers increase voltage and decrease current, reducing I²R losses in power transmission lines.

对于理想变压器，原、副线圈电压比等于匝数比：Vₚ / Vₛ = Nₚ / Nₛ。假设效率100%，输入功率等于输出功率：Vₚ Iₚ = Vₛ Iₛ，因而 Iₚ / Iₛ = Nₛ / Nₚ。升压变压器升高电压、减小电流，从而降低输电线路上的 I²R 损耗。

Alternating current is typically described by root‑mean‑square (rms) values: Iᵣₘₛ = I₀ / √2 and Vᵣₘₛ = V₀ / √2. Most a.c. meters display rms values. In CIE and IB calculations, always check whether peak or rms is required.

交流电通常用方均根值（rms）描述：Iᵣₘₛ = I₀ / √2，Vᵣₘₛ = V₀ / √2。大多数交流电表显示的是有效值。在 CIE 和 IB 的计算题中，务必确认题目要求峰值还是有效值。

10. Summary and Exam Tips | 总结与应试技巧

Electricity and magnetism questions often combine more than one concept — a circuit may involve internal resistance and Kirchhoff’s laws, while an induction question may test both Faraday’s and Lenz’s laws together. Keep these pointers in mind:

电与磁的题目常综合多个概念——一个电路可能涉及内阻和基尔霍夫定律，一道感应题可能同时考查法拉第定律和楞次定律。请牢记以下几点：

• Draw a clear, labelled diagram for any field or circuit problem — it saves time and reduces errors.
• 对于任何场或电路问题，画一个清晰的示意图并标清符号——这能节省时间、减少错误。

• Always state Lenz’s law when explaining the direction of an induced current: “to oppose the change causing it”.
• 解释感应电流方向时，务必使用楞次定律：“感应电流的方向总是阻碍引起它的变化”。

• Use Fleming’s left‑hand rule for motor effect (force on current), and the right‑hand grip rule for generated magnetic fields. Do not confuse them.
• 电动机效应（通电导体受力）用左手定则，电流产生磁场用右手螺旋定则，切勿混淆。

• Check whether the question asks for peak or rms values in a.c. calculations; quoting the wrong one loses marks.
• 交流电计算时仔细辨别题目要求的是峰值还是有效值；答错会丢分。

• Memorise the main formula forms: V = IR, P = VI = I²R, F = BIL sinθ, ε = –N ΔΦ/Δt, Vₚ/Vₛ = Nₚ/Nₛ, and the relationships for series and parallel resistors.
• 熟记核心公式的形式：V = IR，P = VI = I²R，F = BIL sinθ，ε = –N ΔΦ/Δt，Vₚ/Vₛ = Nₚ/Nₛ，以及串、并联电阻的关系。

Master these fundamentals, practise applying them to varied contexts, and electricity and magnetism will become one of your strongest sections in the exam.

掌握这些基础要点，并在不同情境中反复练习应用，电与磁必将成为你考试中最有把握的板块之一。

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